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The nature of the steady state in models of optimal growth under uncertainty

  • Tapan Mitra

    ()

  • Luigi Montrucchio

    ()

  • Fabio Privileggi

    ()

We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent $\alpha $ . Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters $\alpha $ and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever $\alpha < 1/2$ . More delicate is the case $\alpha > 1/2$ . Singularity with respect to Lebesgue measure also appears for values $\alpha ,p$ such that $\alpha < p^{p}\left( 1-p\right)^{\left( 1-p\right) }$ . For $\alpha > p^{p}\left( 1-p\right) ^{\left( 1-p\right) }$ and $1/3\leq p\leq 2/3,$ Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs $\left( \alpha ,p\right) $ . Copyright Springer-Verlag Berlin/Heidelberg 2003

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File URL: http://hdl.handle.net/10.1007/s00199-002-0340-5
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Article provided by Springer in its journal Economic Theory.

Volume (Year): 23 (2003)
Issue (Month): 1 (December)
Pages: 39-71

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Handle: RePEc:spr:joecth:v:23:y:2003:i:1:p:39-71
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  1. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-406, November.
  2. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
  3. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
  4. Mirman, Leonard J, 1972. "On the Existence of Steady State Measures for One Sector Growth Models with Uncertain Technology," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 13(2), pages 271-86, June.
  5. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, vol. 4(3), pages 479-513, June.
  6. Mirman, Leonard J., 1973. "The steady state behavior of a class of one sector growth models with uncertain technology," Journal of Economic Theory, Elsevier, vol. 6(3), pages 219-242, June.
  7. Donaldson, John B. & Mehra, Rajnish, 1983. "Stochastic growth with correlated production shocks," Journal of Economic Theory, Elsevier, vol. 29(2), pages 282-312, April.
  8. Brock, W. A. & Majumdar, M., 1978. "Global asymptotic stability results for multisector models of optimal growth under uncertainty when future utilities are discounted," Journal of Economic Theory, Elsevier, vol. 18(2), pages 225-243, August.
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